Ground state entanglement in one-dimensional translationally invariant quantum systems

We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians (Hn) for the infinite chain. The spectral gap of Hn is Ω(1/poly(n)). Moreover, for any state in the ground space of Hn and any m, there are regions of size m with entanglement entropy Ω(min(m,n)). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings ['An area law for one dimensional quantum systems', J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/Δ, where Δ is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/Δ. Previously, the best known such bound was logarithmic in 1/Δ.